Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 8.9s

Alternatives: 12

Speedup: 1.0×

Specification

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Initial Program: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - t\_0, u, t\_0\right)} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (- (/ 1.0 (fma (- (/ 1.0 (+ 1.0 (exp (- (/ PI s))))) t_0) u t_0)) 1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / fmaf(((1.0f / (1.0f + expf(-(((float) M_PI) / s)))) - t_0), u, t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / fma(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(-Float32(Float32(pi) / s))))) - t_0), u, t_0)) - Float32(1.0))))
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{\mathsf{fma}\left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, u, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]
4. Add Preprocessing

Alternative 2: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{e^{-\frac{\pi}{s}} - -1} - \frac{u}{e^{\frac{\pi}{s}} - -1}} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/ 1.0 (- (/ u (- (exp (- (/ PI s))) -1.0)) (/ u (- (exp (/ PI s)) -1.0))))
    1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / ((u / (expf(-(((float) M_PI) / s)) - -1.0f)) - (u / (expf((((float) M_PI) / s)) - -1.0f)))) - 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u / Float32(exp(Float32(-Float32(Float32(pi) / s))) - Float32(-1.0))) - Float32(u / Float32(exp(Float32(Float32(pi) / s)) - Float32(-1.0))))) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((u / (exp(-(single(pi) / s)) - single(-1.0))) - (u / (exp((single(pi) / s)) - single(-1.0))))) - single(1.0)));
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{\mathsf{fma}\left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, u, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]

4. Taylor expanded in u around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]

6. Applied rewrites 97.6%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]

8. Applied rewrites 97.6%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\log \left(\frac{1}{\frac{u}{e^{-\frac{\pi}{s}} - -1} + \frac{u}{-1 - e^{\frac{\pi}{s}}}} - 1\right)} \]

10. Applied rewrites 97.6%

    \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{\frac{u}{e^{-\frac{\pi}{s}} - -1} - \frac{u}{e^{\frac{\pi}{s}} - -1}}} - 1\right) \]
11. Add Preprocessing

Alternative 3: 94.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{2 + \frac{\pi}{s}}\right)} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (* u (- (/ 1.0 (+ 1.0 (exp (- (/ PI s))))) (/ 1.0 (+ 2.0 (/ PI s))))))
    1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / (u * ((1.0f / (1.0f + expf(-(((float) M_PI) / s)))) - (1.0f / (2.0f + (((float) M_PI) / s)))))) - 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(-Float32(Float32(pi) / s))))) - Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(pi) / s)))))) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / (u * ((single(1.0) / (single(1.0) + exp(-(single(pi) / s)))) - (single(1.0) / (single(2.0) + (single(pi) / s)))))) - single(1.0)));
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{\mathsf{fma}\left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, u, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]

4. Taylor expanded in u around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]

6. Applied rewrites 97.6%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]

7. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{2 + \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} - 1\right) \]

9. Applied rewrites 94.1%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{2 + \color{blue}{\frac{\pi}{s}}}\right)} - 1\right) \]
10. Add Preprocessing

Alternative 4: 94.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{e^{-\frac{\pi}{s}} - -1} + \frac{u}{-1 - \left(1 + \frac{\pi}{s}\right)}} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (+ (/ u (- (exp (- (/ PI s))) -1.0)) (/ u (- -1.0 (+ 1.0 (/ PI s))))))
    1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / ((u / (expf(-(((float) M_PI) / s)) - -1.0f)) + (u / (-1.0f - (1.0f + (((float) M_PI) / s)))))) - 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u / Float32(exp(Float32(-Float32(Float32(pi) / s))) - Float32(-1.0))) + Float32(u / Float32(Float32(-1.0) - Float32(Float32(1.0) + Float32(Float32(pi) / s)))))) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((u / (exp(-(single(pi) / s)) - single(-1.0))) + (u / (single(-1.0) - (single(1.0) + (single(pi) / s)))))) - single(1.0)));
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{\mathsf{fma}\left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, u, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]

4. Taylor expanded in u around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]

6. Applied rewrites 97.6%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]

8. Applied rewrites 97.6%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\log \left(\frac{1}{\frac{u}{e^{-\frac{\pi}{s}} - -1} + \frac{u}{-1 - e^{\frac{\pi}{s}}}} - 1\right)} \]

9. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{e^{-\frac{\pi}{s}} - -1} + \frac{u}{-1 - \left(1 + \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right)}} - 1\right) \]

11. Applied rewrites 94.1%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{e^{-\frac{\pi}{s}} - -1} + \frac{u}{-1 - \left(1 + \color{blue}{\frac{\pi}{s}}\right)}} - 1\right) \]
12. Add Preprocessing

Alternative 5: 37.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{0.5 \cdot u + \frac{u}{-1 - e^{\frac{\pi}{s}}}} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (- (/ 1.0 (+ (* 0.5 u) (/ u (- -1.0 (exp (/ PI s)))))) 1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / ((0.5f * u) + (u / (-1.0f - expf((((float) M_PI) / s)))))) - 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(0.5) * u) + Float32(u / Float32(Float32(-1.0) - exp(Float32(Float32(pi) / s)))))) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((single(0.5) * u) + (u / (single(-1.0) - exp((single(pi) / s)))))) - single(1.0)));
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{\mathsf{fma}\left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, u, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]

4. Taylor expanded in u around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} - 1\right) \]

6. Applied rewrites 97.6%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]

8. Applied rewrites 97.6%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\log \left(\frac{1}{\frac{u}{e^{-\frac{\pi}{s}} - -1} + \frac{u}{-1 - e^{\frac{\pi}{s}}}} - 1\right)} \]

9. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{1}{2} \cdot u + \frac{\color{blue}{u}}{-1 - e^{\frac{\pi}{s}}}} - 1\right) \]

11. Applied rewrites 37.0%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{0.5 \cdot u + \frac{\color{blue}{u}}{-1 - e^{\frac{\pi}{s}}}} - 1\right) \]
12. Add Preprocessing

Alternative 6: 24.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right) - -0.25 \cdot \pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+ 1.0 (* 4.0 (/ (- (* u (- (* -0.25 PI) (* 0.25 PI))) (* -0.25 PI)) s))))))

C

float code(float u, float s) {
	return -s * logf((1.0f + (4.0f * (((u * ((-0.25f * ((float) M_PI)) - (0.25f * ((float) M_PI)))) - (-0.25f * ((float) M_PI))) / s))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(1.0) + Float32(Float32(4.0) * Float32(Float32(Float32(u * Float32(Float32(Float32(-0.25) * Float32(pi)) - Float32(Float32(0.25) * Float32(pi)))) - Float32(Float32(-0.25) * Float32(pi))) / s)))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(1.0) + (single(4.0) * (((u * ((single(-0.25) * single(pi)) - (single(0.25) * single(pi)))) - (single(-0.25) * single(pi))) / s))));
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

2. Taylor expanded in s around -inf

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]

4. Applied rewrites 24.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right) - -0.25 \cdot \pi}{s}\right)} \]
5. Add Preprocessing

Alternative 7: 14.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{s \cdot s}{-0.5 \cdot \pi}}{u} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (/ (/ (* s s) (* -0.5 PI)) u))

C

float code(float u, float s) {
	return ((s * s) / (-0.5f * ((float) M_PI))) / u;
}

Julia

function code(u, s)
	return Float32(Float32(Float32(s * s) / Float32(Float32(-0.5) * Float32(pi))) / u)
end

MATLAB

function tmp = code(u, s)
	tmp = ((s * s) / (single(-0.5) * single(pi))) / u;
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{\mathsf{fma}\left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, u, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]

4. Taylor expanded in u around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{s}{u \cdot \left(\frac{1}{1 + e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} \]

6. Applied rewrites 17.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{s}{u \cdot \left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} \]

7. Taylor expanded in s around -inf

   \[\leadsto \frac{{s}^{2}}{\color{blue}{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}} \]

9. Applied rewrites 14.4%

   \[\leadsto \frac{{s}^{2}}{\color{blue}{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)}} \]

11. Applied rewrites 14.4%

    \[\leadsto \frac{\frac{s \cdot s}{-0.5 \cdot \pi}}{u} \]
12. Add Preprocessing

Alternative 8: 14.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{s \cdot s}{u \cdot \left(-0.5 \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (/ (* s s) (* u (* -0.5 PI))))

C

float code(float u, float s) {
	return (s * s) / (u * (-0.5f * ((float) M_PI)));
}

Julia

function code(u, s)
	return Float32(Float32(s * s) / Float32(u * Float32(Float32(-0.5) * Float32(pi))))
end

MATLAB

function tmp = code(u, s)
	tmp = (s * s) / (u * (single(-0.5) * single(pi)));
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{\mathsf{fma}\left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, u, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]

4. Taylor expanded in u around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{s}{u \cdot \left(\frac{1}{1 + e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} \]

6. Applied rewrites 17.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{s}{u \cdot \left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} \]

7. Taylor expanded in s around -inf

   \[\leadsto \frac{{s}^{2}}{\color{blue}{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}} \]

9. Applied rewrites 14.4%

   \[\leadsto \frac{{s}^{2}}{\color{blue}{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)}} \]

11. Applied rewrites 14.4%

    \[\leadsto \frac{s \cdot s}{\color{blue}{u \cdot \left(-0.5 \cdot \pi\right)}} \]
12. Add Preprocessing

Alternative 9: 14.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \frac{s}{u \cdot \left(-0.5 \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s (/ s (* u (* -0.5 PI)))))

C

float code(float u, float s) {
	return s * (s / (u * (-0.5f * ((float) M_PI))));
}

Julia

function code(u, s)
	return Float32(s * Float32(s / Float32(u * Float32(Float32(-0.5) * Float32(pi)))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * (s / (u * (single(-0.5) * single(pi))));
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

3. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{\mathsf{fma}\left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, u, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} - 1\right) \]

4. Taylor expanded in u around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{s}{u \cdot \left(\frac{1}{1 + e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}} \]

6. Applied rewrites 17.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{s}{u \cdot \left(\frac{1}{1 + e^{-\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} \]

7. Taylor expanded in s around -inf

   \[\leadsto \frac{{s}^{2}}{\color{blue}{u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}} \]

9. Applied rewrites 14.4%

   \[\leadsto \frac{{s}^{2}}{\color{blue}{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)}} \]

11. Applied rewrites 14.4%

    \[\leadsto s \cdot \frac{s}{\color{blue}{u \cdot \left(-0.5 \cdot \pi\right)}} \]
12. Add Preprocessing

Alternative 10: 11.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\pi + \pi, u, -\pi\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (fma (+ PI PI) u (- PI)))

C

float code(float u, float s) {
	return fmaf((((float) M_PI) + ((float) M_PI)), u, -((float) M_PI));
}

Julia

function code(u, s)
	return fma(Float32(Float32(pi) + Float32(pi)), u, Float32(-Float32(pi)))
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

4. Applied rewrites 11.6%

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right)} \]

6. Applied rewrites 11.6%

   \[\leadsto \left(\pi \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)\right) \cdot \color{blue}{4} \]

8. Applied rewrites 11.6%

   \[\leadsto \mathsf{fma}\left(\pi + \pi, \color{blue}{u}, -\pi\right) \]
9. Add Preprocessing

Alternative 11: 11.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \left(\pi + \pi\right) \cdot u - \pi \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- (* (+ PI PI) u) PI))

C

float code(float u, float s) {
	return ((((float) M_PI) + ((float) M_PI)) * u) - ((float) M_PI);
}

Julia

function code(u, s)
	return Float32(Float32(Float32(Float32(pi) + Float32(pi)) * u) - Float32(pi))
end

MATLAB

function tmp = code(u, s)
	tmp = ((single(pi) + single(pi)) * u) - single(pi);
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

4. Applied rewrites 11.6%

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right)} \]

6. Applied rewrites 11.6%

   \[\leadsto \left(\pi \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)\right) \cdot \color{blue}{4} \]

8. Applied rewrites 11.6%

   \[\leadsto \left(\pi + \pi\right) \cdot u - \color{blue}{\pi} \]
9. Add Preprocessing

Alternative 12: 11.4% accurate, 46.3× speedup

Math

\[\begin{array}{l} \\ -\pi \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- PI))

C

float code(float u, float s) {
	return -((float) M_PI);
}

Julia

function code(u, s)
	return Float32(-Float32(pi))
end

MATLAB

function tmp = code(u, s)
	tmp = -single(pi);
end

Derivation

1. Initial program 98.9%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

2. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]

4. Applied rewrites 11.4%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]

6. Applied rewrites 11.4%

   \[\leadsto -\pi \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025161 
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0)) (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (- (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) 1.0))))